The present invention relates to Microelectromechanical (MEMS) devices and, more particularly, to an efficient method for extracting the pull-in instability parameters of electrostatically actuated MEMS devices.
Electrostatic actuation is widely used in MEMS devices to deform elastic elements. The electromechanical response of these actuators may exhibit an inherent instability, shown as the pull-in phenomenon. A voltage difference is applied across the electrodes of an actuator to generate an electrostatic force that tends to reduce the gap between the electrodes. For a sufficiently low voltage, the electrostatic force is balanced by an elastic restoring force. In this stable state the gap between the electrodes is inversely proportional to the applied voltage. Above a certain voltage, the electrostatic force is larger than the restoring elastic force for any deformation. As a result, the actuator is unstable and the gap between the two electrodes rapidly vanishes. The voltage and deformation at the onset of instability are termed pull-in voltage and pull-in deformation, or in short the pull-in parameters of the actuator.
Characterization of the pull-in parameters is important when designing electrostatically actuated micromachined devices. In switching applications, either optical (Hornbeck, U.S. Pat. No. 5,061,049) or electrical (C. T. C. Nguen et al., “Micromachined devices for wireless communications”, Proc. IEEE vol. 86 no. 8 pp. 1756-1768 (August 1998)), the pull-in voltage is minimized to obtain optimal performance. In analog scanning micromirror applications (D. L. Dickensheets and R. G. Kino, “Silicon-micromachined scanning confocal optical microscope”, JMEMS vol. 7 no. 1 pp. 38-47 (March 1998)), the travel range of the actuator is important and therefore the pull-in deformation should be maximized. Modeling tools that can simulate the nonlinear electromechanical response and extract the pull-in parameters of electrostatic actuators arc therefore of great importance. To enable an accurate determination of optimal material and geometrical parameters of actuators, these modeling tools should be based on accurate and efficient calculations.
Several approaches for extracting the pull-in parameters have been reported in the technical literature (P. Osterberg et al., “Self-consistent simulation and modeling of electrostatically deformed diaphragm”, Proc. IEEE MEMS 94, Oiso, pp. 28-32 (January 1994); S. D. Senturia, “CAD challenges for microsensors, microactuators and Microsystems”, Proc. IEEE vol. 86 no. 8 pp. 1611-1626 (1998)) and have been implemented in MEMS CAD tools that are available commercially, for example from Coventor, Inc. of Cary N.C. USA and from Corning Intellisense of Wilmington Mass. USA. Approximate analytical models have been suggested for electrostatic actuators (Y. Nemirovsky and O. Degani, “A methodology and model for the pull-in parameters of electrostatic actuators”, JMEMS vol. 10 no. 4 pp. 601-605 (December 2001): also see S. D. Senturia, Microsystem Design, Kluwer Academic Press, Boston, 2001). These models yield fast results but are limited to actuators with very few degrees of freedom. To accurately calculate the pull-in parameters of general deformable elements, such as beam and plate actuators, that have (in the continuum limit) an infinite number of degrees of freedom, a more general approach has been suggested (E. K. Chan et al., “Characterization of contact electromechanics through capacitance-voltage measurements and simulations”, JMEMS vol. 8 no. 2 pp. 208-217 (June 1999); R. K. Gupta, Electrostatic Pull-In Test Structure Design for In-Situ Mechanical Property Measurements of Microelectromechanical Systems (MEMS), PhD Thesis, Massachusetts Institute of Technology, June 1997). In this approach, which is referred to herein as the voltage-iteration (VI) method, the electromechanical response of the actuator is numerically simulated by fixing the applied voltage. The pull-in parameters are calculated by iteratively approaching the pull-in voltage with decreasing voltage increments. This algorithm has been implemented in a finite-difference scheme and in coupled finite-elements (FEM) and boundary-elements (BEM) scheme. See Osterberg et al., and also M. Fischer et al., “electrostatically deflectable polysilicon micromirrors—dynamic behaviour and comparison with results from FEM modeling with ANSYS, Sensors and Actuators A vol. 67 pp. 89-95 (1998).
A typical static equilibrium curve of an electrostatic actuator is schematically depicted in FIG. 1. The convex function describes the applied voltage as function of a representative parameter of the actuator deformation, for example, the displacement of the center of a clamped-clamped beam. For deformations smaller than the pull-in deformation, the static equilibrium state is stable (solid line). In contrast, for deformations larger than the pull-in deformation the static equilibrium state is unstable (dashed line).
Two aspects of the physical response of electrostatic actuators are apparent in FIG. 1. First, the voltage is a unique function of the deformation, whereas the deformation is not a unique function of the voltage. Second, the maximal deformation can be trivially estimated, as it is bounded from above by the gap between the electrodes. In contrast, the maximal voltage cannot be estimated a priori.
In the VI algorithm, the pull-in voltage is approached iteratively. At each iteration, the static equilibrium deformation is calculated for an applied voltage. This calculation can be carried out by a relaxation method, a Newton-Raphson method, or a host of other numerical schemes. If the deformation calculation converges, it is concluded that the applied voltage is below the pull-in value. On the other hand, if the calculated deformation fails to converge it is concluded that the applied voltage is higher than the pull-in value. Several methods have been employed in the references cited above to establish whether the deformation calculation converges. The interval between these two limits is continuously decreased until the voltage interval is smaller than a predetermined accuracy. The iterations are represented by the set of horizontal dashed lines in FIG. 1. It can easily be seen that not all the horizontal lines cross the equilibrium curve, and therefore not all lines are associated with equilibrium states.
FIG. 2 is a flow chart of the VI algorithm. In block 102, an initial trial value of an applied voltage V is selected. In block 104, an attempt is made to calculate the deformation corresponding to the present value of V. If the deformation calculation does not converge (block 106), then V is decreased (block 108) and the deformation calculation is attempted again in block 104. If the deformation calculation does converge (block 106), and V is less than its previously established upper bound by less than a predetermined accuracy (block 110) then the present value of V is taken as an estimate of the pull-in voltage (block 114). Otherwise, V is increased (block 112) and the deformation calculation is attempted again in block 104.
The main advantage of the VI algorithm is its simplicity and ease of integration into commercial CAD tools. For any applied voltage, the electro-elastic problem is solved by iteratively solving uncoupled electrostatic and elastic problems. It is therefore easy to implement this algorithm by sequentially employing existing numerical codes for each of these problems.